| This thesis, written under the supervision of W. Schmid, provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed in [Sch].;A corresponding problem in the compact group setting was solved by N. Berline, E. Getzler and M. Vergne in [BGV] by an application of the theory of equivariant forms and particularly the fixed point integral localization formula. This thesis (besides its representation-theoretical significance) provides a whole family of examples where it is possible to localize integrals to fixed points with respect to an action of a noncompact group. Moreover, a localization argument given here is not specific to the particular setting considered in this thesis and can be extended to a more general situation.;We start with the right hand side of the integral character formula gR q4=12p inn! ChFm *l4&d4; -s+p*tl n and show that it is equivalent to the right hand side of the fixed point character formula gR q4=gR k=1 Wmxk ge&angl0;g,lx kg&angr0; axkg ,1g&ldots;a xkg,n g4g . .;We observe that the integrand in the integral character formula is a closed form. We use the open embedding theorem of W. Schmid and K. Vilonen ([SchV1]) to construct a deformation of Ch( F ) into a cycle of a very simple kind such that the integral stays unchanged in the process of deformation.;For the purpose of making our integrand an L 1-object, we introduce another deformation thetat( g) : T*X → T*X, where t ∈ [0, 1] and g lies in a certain subset g'R⊂g R whose complement has measure zero. Lemma 17 says that theta t(g) also leaves the integral unchanged.;The key ideas are the deformation of Ch( F ), the definition of thetat( g) : T*X → T*X and Lemma 17. Because of the right definition of thetat( g), Lemma 17 holds and our calculation of the integral becomes very simple. We see that, as t → 0+, the integral concentrates more and more inside T*U, where U is a neighborhood of the set of fixed points of g in the flag variety X. In the limit, we obtain the right hand side of the fixed point character formula. This means that the integral is localized at the fixed points of g. |
